July 1, 2015

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- There is a strong temporal
**trend**that needs to be accounted for

- Family of
**orthogonal**polynomials**dense**in \(\mathscr{L}_2\) - Any regular curve can be approximated as much as needed by taking a linear combination of polynomials up to a high-enough order

\[ \begin{aligned} y_{ij} = & \mathrm{X}_i \beta + \Sigma_{k = 0}^{\mathrm{ord}} a_{ik} \mathcal{L}_k(j) + \varepsilon_{ij} \\ [a_0', \ldots, a_{\mathrm{ord}}']' \sim & N(0, \Sigma_{\mathrm{ord}} \otimes \mathrm{A})\\ \varepsilon \sim & N(0, \sigma_e^2) \end{aligned} \]

The Breeding Value of an individual is a

**function**of an environmental variable (temp., precip., …)This function is parameterised as a

**linear combination**of Legendre orthogonal polynomials of order up to a given`order`

Each individual is described by

`order`

+ 1**coefficients**

- Functional Breeding Values for each individual